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line-to-line voltage in rms value
. 
We follow this convention unless
 
stated otherwise in the remaining 
portions of this chapter. Thus a 400 V, 50 Hz source means a source that has a line-to-line voltage 
magnitude of 400 V rms at 50 Hz. If it is Y-connected its phase voltage will be 400/
√
3 V rms and peak 
of its phase voltage will be 400
√
2/
√
3 V.
8.3 
analysIs of Balanced three-Phase cIrcuIts
Balanced three-phase circuits comprising a balanced three-phase source and a balanced three-phase 
load can be of four varieties, viz., –Y–Y, Y–
D
, 
D
–Y or 
D
–
D
. We develop a common analysis procedure 
for all the possible configurations.
8.3.1 
equivalence Between a y-connected source and a 
D
-connected source
Let 
V
RY
=
V
L
∠
0
°
, 
V
YB
=
V
L
∠-
120
°
and 
V
BR
=
V
L
∠
120
°
be the three line voltages observed 
in a three-phase circuit and let 
I
R
=
I
L
∠-
(
q
+ 
30
°
), 
I
Y
=
I
L
∠-
(
q
+ 
150
°
) and 
I
B
=
I
L
∠-
(
q
-
90
°
) 
be the observed line currents where the observed phase lag of first line current is expressed 
as 30
°
plus some angle designated by 
q
. If the source is within a black box and we are to guess 
whether it is a Y-connected source or a 
D
-connected source, we will find that we will not be able 
to resolve the matter using the observed line voltage phasors and line current phasors. This is so 
because both configurations shown in Fig. 8.3-1 will result in these observed line voltages and line
currents.
V
L
∠
–120°
∠
–
–
∠
0º
V
L
∠
120°
=
=
∠
–(
+30°)
–
+
–
–
–
–
–
+
+
–
–
–
+
+
+
+
+
N
B
+
Y
B